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Chapter 12: Problem 32

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{\sqrt{3}}$$

Short Answer

Expert verified
\(-0.5\)

Step by step solution

01

Understanding Logarithmic Expression

The expression \(\log_3 \frac{1}{\sqrt{3}}\) can be broken down with the understanding that the logarithm base 3 of 1 divided by the square root of 3 is the power to which 3 must be raised to get \(\frac{1}{\sqrt{3}}\).
02

Simplifying Numerator

Notice that \(\frac{1}{\sqrt{3}}\) can be simplified. We have to rationalize the denominator by multiplying both numerator and denominator by \(\sqrt{3}\), which yields \(\frac{\sqrt{3}}{3}\). Now our expression becomes \(\log_3 \frac{\sqrt{3}}{3}\).
03

Using Properties of Logarithms

The expression can now be rewritten into subtraction form utilizing logarithmic rules. According to the rules, \(\log_b \frac{m}{n}= \log_b m - \log_b n\). It then becomes \(\log_3 {\sqrt{3}} - \log_3 3\).
04

Calculate Individual Logarithms

Finally, the log base 3 of square root of 3 is equal to \(0.5\), because 3 raised to power \(0.5\) equals to square root of 3. And log base 3 of 3 is equal to \(1\), because 3 raised to power 1 is 3. Our final expression is therefore \(0.5 - 1\).
05

Final Calculation

Subtracting 1 from 0.5 results to \(-0.5\). That's the final answer.

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Most popular questions from this chapter

Solve each equation. $$3^{x+2} \cdot 3^{x}=81$$

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the \(\operatorname{graph}\) of \(g(x)=\log _{2} x\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\ln \sqrt{2}=\frac{\ln 2}{2}$$

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}, \ldots\) Describe what you observe.

Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)
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